For 1 ≤ q ≤ α ≤ p ≤ ∞, ${(L^q,l^p)}^{\alpha }$ is a complex Banach space which is continuously included in the Wiener amalgam space $(L^q,l^p)$ and contains the Lebesgue space $L^{\alpha }$. We study the closure ${(L^q,l^p)}_{c,0}^{\alpha }$ in ${(L^q,l^p)}^{\alpha }$ of the space of test functions (infinitely differentiable and with compact support in ${ℝ}^d$) and obtain norm inequalities for Riesz potential operators and Riesz transforms in these spaces. We also introduce the Sobolev type space $W¹\left({(L^q,l^p)}^{\alpha }\right)$ (a subspace of a Morrey-Sobolev space, but a superspace of the classical Sobolev space...

Let 1 ≤ p < ∞, k ≥ 1, and let Ω ⊂ ℝⁿ be an arbitrary open set. We prove a converse of the Calderón-Zygmund theorem that a function $f\in W^{k,p}\left(\Omega \right)$ possesses an $L^p$ derivative of order k at almost every point x ∈ Ω and obtain a characterization of the space $W^{k,p}\left(\Omega \right)$. Our method is based on distributional arguments and a pointwise inequality due to Bojarski and Hajłasz.

Let $(X,d,\mu )$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator of order $m$ on $L^2\left(X\right)$. Assume that the semigroup ${\mathrm{e}}^{-tL}$ generated by $L$ satisfies the Davies-Gaffney estimate of order $m$ and $L$ satisfies the Plancherel type estimate. Let $H_L^p\left(X\right)$ be the Hardy space associated with $L.$ We show the boundedness of Stein’s square function ${𝒢}_{\delta }\left(L\right)$ arising from Bochner-Riesz means associated to $L$ from Hardy spaces $H_L^p\left(X\right)$ to $L^p\left(X\right)$, and also study...

For $1\le n<d$ integers and $\rho >2$, we prove that an $n$-dimensional Ahlfors-David regular measure $\mu$ in ${ℝ}^d$ is uniformly $n$-rectifiable if and only if the $\rho$-variation for the Riesz transform with respect to $\mu$ is a bounded operator in $L^2\left(\mu \right)$. This result can be considered as a partial solution to a well known open problem posed by G. David and S. Semmes which relates the $L^2\left(\mu \right)$ boundedness of the Riesz transform to the uniform rectifiability of $\mu$.

Let ${ℒ}_1=-\Delta +V$ be a Schrödinger operator and let ${ℒ}_2={(-\Delta )}^2+V^2$ be a Schrödinger type operator on ${ℝ}^n$ $(n\ge 5)$, where $V\ne 0$ is a nonnegative potential belonging to certain reverse Hölder class $B_s$ for $s\ge n/2$. The Hardy type space $H_{{ℒ}_2}^1$ is defined in terms of the maximal function with respect to the semigroup $\left\{{\mathrm{e}}^{-t{ℒ}_2}\right\}$ and it is identical to the Hardy space $H_{{ℒ}_1}^1$ established by Dziubański and Zienkiewicz. In this article, we prove the $L^p$-boundedness of the commutator ${ℛ}_b=bℛf-ℛ\left(bf\right)$ generated by the Riesz transform $ℛ={\nabla }^2{ℒ}_2^{-1/2}$, where $b\in {\mathrm{BMO}}_{\theta }\left(\rho \right)$, which is larger...

Let $L$ be a second order divergence form elliptic operator with complex bounded measurable coefficients. The operators arising in connection with $L$, such as the heat semigroup and Riesz transform, are not, in general, of Calderón-Zygmund type and exhibit behavior different from their counterparts built upon the Laplacian. The current paper aims at a thorough description of the properties of such operators in $L^p$, Sobolev, and some new Hardy spaces naturally associated to $L$. First, we show...

We define a new function space $B$, which contains in particular BMO, BV, and $W^{1/p,p}$, $1<p<\infty$. We investigate its embedding into Lebesgue and Marcinkiewicz spaces. We present several inequalities involving $L^p$ norms of integer-valued functions in $B$. We introduce a significant closed subspace, $B_0$, of $B$, containing in particular VMO and $W^{1/p,p}$, $1\le p<\infty$. The above mentioned estimates imply in particular that integer-valued functions belonging to $B_0$ are necessarily constant. This framework provides a “common roof”...

We obtain the boundedness of Calderón-Zygmund singular integral operators $T$ of non-convolution type on Hardy spaces $H^p\left(𝒳\right)$ for $1/(1+ϵ)<p\le 1$, where $𝒳$ is a space of homogeneous type in the sense of Coifman and Weiss (1971), and $ϵ$ is the regularity exponent of the kernel of the singular integral operator $T$. Our approach relies on the discrete Littlewood-Paley-Stein theory and discrete Calderón’s identity. The crucial feature of our proof is to avoid atomic decomposition and molecular theory in contrast...

Let f be a measurable function defined on ℝ. For each n ∈ ℤ we consider the average $Aₙf\left(x\right)=2^{-n}{\int }_x^{x+2ⁿ}f$. The square function is defined as $Sf\left(x\right)=({\sum }_{n=-\infty }^{\infty }|Aₙf\left(x\right)-A_{n-1}f\left(x\right){\left|²\right)}^{1/2}$. The local version of this operator, namely the operator $S₁f\left(x\right)=({\sum }_{n=-\infty }^0|Aₙf\left(x\right)-A_{n-1}f\left(x\right){\left|²\right)}^{1/2}$, is of interest in ergodic theory and it has been extensively studied. In particular it has been proved [3] that it is of weak type (1,1), maps $L^p$ into itself (p > 1) and $L^{\infty }$ into BMO. We prove that the operator S not only maps $L^{\infty }$ into BMO but it also maps BMO into BMO. We also prove that the $L^p$ boundedness...

We prove an interpolatory estimate linking the directional Haar projection $P^{\left(\epsilon \right)}$ to the Riesz transform in the context of Bochner-Lebesgue spaces $L^p(ℝⁿ;X)$, 1 < p < ∞, provided X is a UMD-space. If ${\epsilon }_{i₀}=1$, the result is the inequality $\left|\right|P^{\left(\epsilon \right)}{u\left|\right|}_{L^p(ℝⁿ;X)}{\le C\left|\right|u\left|\right|}_{L^p(ℝⁿ;X)}^{1/}\left|\right|R_{i₀}u{\left|\right|}_{L^p(ℝⁿ;X)}^{1-1/}$, (1) where the constant C depends only on n, p, the UMD-constant of X and the Rademacher type of $L^p(ℝⁿ;X)$. In order to obtain the interpolatory result (1) we analyze stripe operators $S_{\lambda }$, λ ≥ 0, which are used as basic building blocks to dominate the directional Haar projection....

For a function $f\in L_{loc}^p\left(ℝⁿ\right)$ the notion of p-mean variation of order 1, ${₁}^p(f,ℝⁿ)$ is defined. It generalizes the concept of F. Riesz variation of functions on the real line ℝ¹ to ℝⁿ, n > 1. The characterisation of the Sobolev space $W^{1,p}\left(ℝⁿ\right)$ in terms of ${₁}^p(f,ℝⁿ)$ is directly related to the characterisation of $W^{1,p}\left(ℝⁿ\right)$ by Lipschitz type pointwise inequalities of Bojarski, Hajłasz and Strzelecki and to the Bourgain-Brezis-Mironescu approach.

We prove that the Hausdorff operator generated by a function ϕ is bounded on the real Hardy space $H^p\left(ℝ\right)$, 0 < p ≤ 1, if the Fourier transform ϕ̂ of ϕ satisfies certain smoothness conditions. As a special case, we obtain the boundedness of the Cesàro operator of order α on $H^p\left(ℝ\right)$, 2/(2α+1) < p ≤ 1. Our proof is based on the atomic decomposition and molecular characterization of $H^p\left(ℝ\right)$.

Let ${(X,\varrho ,\mu )}_{d,\theta }$ be a space of homogeneous type, i.e. X is a set, ϱ is a quasi-metric on X with the property that there are constants θ ∈ (0,1] and C₀ > 0 such that for all x,x’,y ∈ X, $|\varrho (x,y)-\varrho (x^{\text{'}},y)|\le C₀\varrho {(x,x^{\text{'}})}^{\theta }{[\varrho (x,y)+\varrho (x^{\text{'}},y)]}^{1-\theta }$, and μ is a nonnegative Borel regular measure on X such that for some d > 0 and all x ∈ X, $\mu \left(y\in X:\varrho (x,y)<r\right)\sim r^d$. Let ε ∈ (0,θ], |s| < ε and maxd/(d+ε),d/(d+s+ε) < q ≤ ∞. The author introduces new inhomogeneous Triebel-Lizorkin spaces $F_{\infty q}^s\left(X\right)$ and establishes their frame characterizations by first establishing a Plancherel-Pólya-type...

We study embeddings of spaces of Besov-Morrey type, $i{d}_{\Omega }{:}_{p₁,u₁,q₁}^{s₁}\left(\Omega \right){\hookrightarrow }_{p₂,u₂,q₂}^{s₂}\left(\Omega \right)$, where $\Omega \subset {ℝ}^d$ is a bounded domain, and obtain necessary and sufficient conditions for the continuity and compactness of $i{d}_{\Omega }$. This continues our earlier studies relating to the case of ${ℝ}^d$. Moreover, we also characterise embeddings into the scale of $L_p$ spaces or into the space of bounded continuous functions.

We establish the following sharp local estimate for the family ${R_j}_{j=1}^d$ of Riesz transforms on ${ℝ}^d$. For any Borel subset A of ${ℝ}^d$ and any function $f:{ℝ}^d\to ℝ$, ${\int }_A|R_{j}f\left(x\right)|dx\le C_p{\left|\right|f\left|\right|}_{L^p\left({ℝ}^d\right)}{\left|A\right|}^{1/q}$, 1 < p < ∞. Here q = p/(p-1) is the harmonic conjugate to p, $C_p={[2^{q+2}\Gamma (q+1)/{\pi }^{q+1}{\sum }_{k=0}^{\infty }{(-1)}^k/{(2k+1)}^{q+1}]}^{1/q}$, 1 < p < 2, and $C_p={[4\Gamma (q+1)/{\pi }^q{\sum }_{k=0}^{\infty }1/{(2k+1)}^q]}^{1/q}$, 2 ≤ p < ∞. This enables us to determine the precise values of the weak-type constants for Riesz transforms for 1 < p < ∞. The proof rests on appropriate martingale inequalities, which are of independent interest.

We investigate the construction of Carleson measures from families of multilinear integral operators applied to tuples of $L^{\infty }$ and BMO functions. We show that if the family $R_t$ of multilinear operators has cancellation in each variable, then for BMO functions b₁, ..., bₘ, the measure $|R_t(b₁,...,bₘ)\left(x\right)|²dxdt/t$ is Carleson. However, if the family of multilinear operators has cancellation in all variables combined, this result is still valid if $b_j$ are $L^{\infty }$ functions, but it may fail if $b_j$ are unbounded BMO functions, as...

We consider functions $u\in W_0^{m,1}\left(\Omega \right)$, where $\Omega \subset {ℝ}^N$ is a smooth bounded domain, and $m\ge 2$ is an integer. For all $j\ge 0,1\le k\le m-1$, such that $1\le j+k\le m$, we prove that $\frac{{\partial }^{i}u\left(x\right)}{d{\left(x\right)}^{m-j-k}}\in W_0^{k,1}\left(\Omega \right)$ with $∥{\partial }^k{\left(\frac{{\partial }^{i}u\left(x\right)}{d{\left(x\right)}^{m-j-k}}\right)}_{L^1\left(\Omega \right)}\le C∥u∥{}_{W^{m,1}\left(\Omega \right)}$, where $d$ is a smooth positive function which coincides with dist$(x,\partial \Omega )$ near $\partial \Omega$, and ${\partial }^l$ denotes any partial differential operator of order $l$.

The main purpose of this paper is to prove that the boundedness of the commutator [...] Mκ,b∗ ${ℳ}_{\kappa ,b}^{*}$ generated by the Littlewood-Paley operator [...] Mκ∗ ${ℳ}_{\kappa }^{*}$ and RBMO (μ) function on non-homogeneous metric measure spaces satisfying the upper doubling and the geometrically doubling conditions. Under the assumption that the kernel of [...] Mκ∗ ${ℳ}_{\kappa }^{*}$ satisfies a certain Hörmander-type condition, the authors prove that [...] Mκ,b∗ ${ℳ}_{\kappa ,b}^{*}$ is bounded on Lebesgue spaces Lp(μ) for 1 < p < ∞, bounded from...

CONTENTSIntroduction.......................................................................................................................................................... 5Chapter I. Some preliminary lemmas............................................................................................................ 8Chapter II. Weighted $H^p$ spaces of analytic functions.......................................................................... 13 1. Behaviour at the boundary..........................................................................................................................

Let Φ be a concave function on (0,∞) of strictly critical lower type index $p_{\Phi }\in (0,1]$ and $\omega \in A_{\infty }^{loc}\left(ℝⁿ\right)$ (the class of local weights introduced by V. S. Rychkov). We introduce the weighted local Orlicz-Hardy space $h_{\omega }^{\Phi }\left(ℝⁿ\right)$ via the local grand maximal function. Let $\rho \left(t\right)\equiv t^{-1}/{\Phi }^{-1}\left(t^{-1}\right)$ for all t ∈ (0,∞). We also introduce the BMO-type space $bm{o}_{\rho ,\omega }\left(ℝⁿ\right)$ and establish the duality between $h_{\omega }^{\Phi }\left(ℝⁿ\right)$ and $bm{o}_{\rho ,\omega }\left(ℝⁿ\right)$. Characterizations of $h_{\omega }^{\Phi }\left(ℝⁿ\right)$, including the atomic characterization, the local vertical and the local nontangential maximal function characterizations, are...

The estimate ${∥D^{k-1}u∥}_{L^{n/(n-1)}}\le {∥A\left(D\right)u∥}_{L^1}$ is shown to hold if and only if $A\left(D\right)$ is elliptic and canceling. Here $A\left(D\right)$ is a homogeneous linear differential operator $A\left(D\right)$ of order $k$ on ${ℝ}^n$ from a vector space $V$ to a vector space $E$. The operator $A\left(D\right)$ is defined to be canceling if ${\bigcap }_{\xi \in {ℝ}^n\setminus \left\{0\right\}}A\left(\xi \right)\left[V\right]=\left\{0\right\}$. This result implies in particular the classical Gagliardo–Nirenberg–Sobolev inequality, the Korn–Sobolev inequality and Hodge–Sobolev estimates for differential forms due to J. Bourgain and H. Brezis. In the proof, the class of cocanceling homogeneous...

Let $\Omega \in L^s\left({\mathrm{S}}^{n-1}\right)$ for $s\ge 1$ be a homogeneous function of degree zero and $b$ a BMO function. The commutator generated by the Marcinkiewicz integral ${\mu }_{\Omega }$ and $b$ is defined by $${\displaystyle [b,{\mu }_{\Omega }]\left(f\right)\left(x\right)=({\int }_0^{\infty }{\left|{\int }_{|x-y|\le t}\frac{\Omega (x-y)}{{|x-y|}^{n-1}}[b\left(x\right)-b\left(y\right)]f\left(y\right)\mathrm{d}y{|}^2\frac{\mathrm{d}t}{t^3}\right)}^{1/2}.}$$ In this paper, the author proves the $(L^{p(·)}\left({ℝ}^n\right),L^{p(·)}\left({ℝ}^n\right))$-boundedness of the Marcinkiewicz integral operator ${\mu }_{\Omega }$ and its commutator $[b,{\mu }_{\Omega }]$ when $p(·)$ satisfies some conditions. Moreover, the author obtains the corresponding result about ${\mu }_{\Omega }$ and $[b,{\mu }_{\Omega }]$ on Herz spaces with variable exponent.